Synthesis of a DNF Formula From a Sample of Strings

Monotone DNF Formula That Has a Minimal or Maximal Number of Satisfying Assignments

Lecture Notes in Computer Science - Computing and Combinatorics ◽

10.1007/978-3-540-69733-6_20 ◽

2008 ◽

pp. 191-203

Author(s):

Takayuki Sato ◽

Kazuyuki Amano ◽

Eiji Takimoto ◽

Akira Maruoka

Keyword(s):

Dnf Formula

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A form of feasible interpolation for constant depth Frege systems

Journal of Symbolic Logic ◽

10.2178/jsl/1268917504 ◽

2010 ◽

Vol 75 (2) ◽

pp. 774-784 ◽

Cited By ~ 4

Author(s):

Jan Krajíček

Keyword(s):

Constant Depth ◽

First Order ◽

Order Language ◽

Frege Systems ◽

Image Position ◽

The Universe ◽

Dnf Formula

AbstractLet L be a first-order language and Φ and Ψ two L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle ChainL,Φ,Ψ(n, m) holds: For any chain of finite L-structures C1, …, Cm with the universe [n] one of the following conditions must fail:For each fixed L and parameters n, m the principle ChainL,Φ,Ψ(n,m) can be encoded into a propositional DNF formula of size polynomial in n, m.For any language L containing only constants and unary predicates we show that there is a constant CL such that the following holds: If a constant depth Frege system in DeMorgan language proves ChainL,Φ,Ψ(n, cL . n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying ψ by a depth 3 formula of size 2log(S)O(1) and with bottom fan-in log(S)O(1).

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The Good Old Davis-Putnam Procedure Helps Counting Models

Journal of Artificial Intelligence Research ◽

10.1613/jair.601 ◽

1999 ◽

Vol 10 ◽

pp. 457-477 ◽

Cited By ~ 36

Author(s):

E. Birnbaum ◽

E. L. Lozinskii

Keyword(s):

Present Knowledge ◽

Worst Case ◽

Running Time ◽

Exact Number ◽

Average Running Time ◽

Practical Performance ◽

Series Of Experiments ◽

Case Based ◽

Cnf Formulas ◽

Dnf Formula

As was shown recently, many important AI problems require counting the number of models of propositional formulas. The problem of counting models of such formulas is, according to present knowledge, computationally intractable in a worst case. Based on the Davis-Putnam procedure, we present an algorithm, CDP, that computes the exact number of models of a propositional CNF or DNF formula F. Let m and n be the number of clauses and variables of F, respectively, and let p denote the probability that a literal l of F occurs in a clause C of F, then the average running time of CDP is shown to be O(nm^d), where d=-1/log(1-p). The practical performance of CDP has been estimated in a series of experiments on a wide variety of CNF formulas.

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Not All FPRASs are Equal: Demystifying FPRASs for DNF-Counting (Extended Abstract)

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2019/865 ◽

2019 ◽

Author(s):

Kuldeep S. Meel ◽

Aditya A. Shrotri ◽

Moshe Y. Vardi

Keyword(s):

Monte Carlo ◽

Empirical Study ◽

Time Complexity ◽

Fundamental Problem ◽

Approximation Schemes ◽

Search Procedure ◽

Number Of Solutions ◽

Monte Carlo Techniques ◽

Dnf Formula

The problem of counting the number of solutions of a DNF formula, also called #DNF, is a fundamental problem in AI with wide-ranging applications. Owing to the intractability of the exact variant, efforts have focused on the design of approximate techniques. Consequently, several Fully Polynomial Randomized Approximation Schemes (FPRASs) based on Monte Carlo techniques have been proposed. Recently, it was discovered that hashing-based techniques too lend themselves to FPRASs for #DNF. Despite significant improvements, the complexity of the hashing-based FPRAS is still worse than that of the best Monte Carlo FPRAS by polylog factors. Two questions were left unanswered in previous works: Can the complexity of the hashing-based techniques be improved? How do these approaches compare empirically? In this paper, we first propose a new search procedure for the hashing-based FPRAS that removes the polylog factors from its time complexity. We then present the first empirical study of runtime behavior of different FPRASs for #DNF, which produces a nuanced picture. We observe that there is no single best algorithm for all formulas and that the algorithm with one of the worst time complexities solves the largest number of benchmarks.

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